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An improvement of Hoffmann-Jorgensen's inequality

Klass, Michael J. and Nowicki, Krzysztof LU (2000) In Annals of Probability 28(2). p.851-862
Abstract
Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set || x || = supΛ∈F Λ (x) (||· || is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n ||Σk j=1 Xj || where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of {|| Xi ||}n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ(|| Sn ||) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp... (More)
Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set || x || = supΛ∈F Λ (x) (||· || is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n ||Σk j=1 Xj || where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of {|| Xi ||}n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ(|| Sn ||) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp bounds of S* n which are within a factor of 4 for all p ≥ 1 and within a factor of 2 in the limit as p → ∞. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
expo- nential inequalities, Tail probability inequalities, Hoffmann-Jorgensen's inequality, Banach space valued random variables
in
Annals of Probability
volume
28
issue
2
pages
851 - 862
publisher
Institute of Mathematical Statistics
ISSN
0091-1798
language
English
LU publication?
yes
id
743fac1c-6501-412a-a36e-3addac2f0c0b (old id 1766847)
date added to LUP
2011-01-27 13:05:09
date last changed
2016-04-16 06:42:21
@misc{743fac1c-6501-412a-a36e-3addac2f0c0b,
  abstract     = {Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set || x || = supΛ∈F Λ (x) (||· || is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n ||Σk j=1 Xj || where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of {|| Xi ||}n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ(|| Sn ||) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp bounds of S* n which are within a factor of 4 for all p ≥ 1 and within a factor of 2 in the limit as p → ∞.},
  author       = {Klass, Michael J. and Nowicki, Krzysztof},
  issn         = {0091-1798},
  keyword      = {expo- nential inequalities,Tail probability inequalities,Hoffmann-Jorgensen's inequality,Banach space valued random variables},
  language     = {eng},
  number       = {2},
  pages        = {851--862},
  publisher    = {ARRAY(0x81af738)},
  series       = {Annals of Probability},
  title        = {An improvement of Hoffmann-Jorgensen's inequality},
  volume       = {28},
  year         = {2000},
}